I am looking into the matrix norm $\|\cdot\|_{p,q}, p,q\ge1$, induced by vector norms. I am having trouble deriving the explicit expression for the matrix norm in terms of the matrix elements.
For matrix $A:=[a_{i,j}]_{i,j}$ and a column matrix $x:=[x_j]_j$, \begin{align} \|Ax\|_q &= \bigg(\sum_i\Big|\sum_j a_{i,j}x_j\Big|^q\bigg)^\frac1q \\ &\le \sum_j \Big(\sum_i |a_{i,j}x_j|^q \Big)^\frac1q \quad \text{ by Minkowski's inequality} \tag1\label1\\ &=\sum_j |x_j|\Big(\sum_i |a_{i,j}|^q\Big)^\frac1q \\ &\le \Big(\sum_j |x_j|^p\Big)^\frac1p\bigg(\sum_j\Big(\sum_i |a_{i,j}|^q\Big)^\frac{p'}q\bigg)^\frac1{p'} \quad\text{by Hölder's inequality} \tag2\label2 \end{align} where $p'$ is the Hölder conjugate so that $\frac1p+\frac1{p'}=1$. So I would think $$\|A\|_{p,q}=\bigg(\sum_j\Big(\sum_i |a_{i,j}|^q\Big)^\frac{p'}q\bigg)^\frac1{p'}. \tag3\label3$$ The part I am not sure is how I would simultaneously make inequalities \eqref{1} and \eqref{2} equalities in order to prove the upper bound of \eqref{2} is reachable.
However, this Wikipedia section indicates that $$\| A \|_{p,q} = \left(\sum_{j=1}^n \left( \sum_{i=1}^m |a_{i,j}|^p \right)^{\frac{q}{p}}\right)^{\frac{1}{q}}$$ quite different from Equation \eqref{3}. Note: As pointed out by @QiaochuYuan in the comments below, this matrix norm notation is overloaded and symbolizes a different norm from the induced matrix norm defined above. We should correct the notation.
What is the correct derivation of $\| A \|_{p,q}$?